Tautological classes on moduli spaces of hyper-Kähler manifolds

Abstract

We study algebraic cycles on moduli spaces ${\mathcal{F}}_h$ of $h$-polarized hyper-Kähler manifolds. Following previous work of Marian, Oprea, and Pandharipande on the tautological conjecture on moduli spaces of K3 surfaces, we first define the tautological ring on ${\mathcal{F}}_h$. We then study the images of these tautological classes in the cohomology groups of ${\mathcal{F}}_h$ and prove that most of them are linear combinations of Noether–Lefschetz cycle classes. In particular, we prove the cohomological version of the tautological conjecture on moduli space of K3${}^{[n]}$-type hyper-Kähler manifolds with $n\le 2$. Secondly, we prove the cohomological generalized Franchetta conjecture on a universal family of these hyper-Kähler manifolds.